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I was wondering what happens if one has a photon that is so energetic that it gets trapped in its own gravitational field.

Imagine a mass m orbitting a large mass M in a circular orbit with radius r and with transverse velocity v. By equating the gravitational force on m with its centripetal acceleration we have:

G M m / r^2 = m v^2 / r

Now let velocity v approach the speed of light c. The mass m will be boosted by a Lorentz factor but I think the equation should still hold so that we have:

G M m / r^2 = m c^2 / r

Thus cancelling m we have:

M = (c^2 / G) r

or in terms of Energy E = M c^2 we have

E = (c^4 / G) r (*)

This is the energy that a photon must have to get trapped in its own gravitational field so

that it orbits in a circle with radius r. The quantitiy c^4 / G is also the string tension so maybe this is also a model of a string.

Now let us suppose that the photon has angular momentum hbar. Then we have

r * p = hbar

where p is the linear momentum of the photon.

Now we know that p = E / c for photons so that we have:

r * E / c = hbar

E = hbar * c / r

E = h * c / 2.pi.r

E = h / t (**)

where t is the time period of the photon orbit.

If we substitute E=h/t into equation (*) we get:

h / t = (c^4 / G) * r

Rearranging we get:

r * t = h G / c^4

Thus we find that the spacetime area occupied by this bound photon, r * t, is quantized.

I was wondering if this is similar to the result that the world sheet area swept out by strings is quantized.

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# Gravitationally trapped photons leads to quantized space-time

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